Sentence examples for proofs can be given from inspiring English sources

The proofs can be given as in the corresponding results in [32, 33].

Though different and even shorter proofs can be given, based on completely different strategies, the one we present here has the advantage of not using any sophisticate technical tool, it is geometrically intuitive and can be adapted to other contexts where symmetry plays a fundamental role.

But the proof can be given as follows.

Using Lemma 3.4 v) instead of (ii), the proof can be given in a similar way.

The achievability proof can be given on similar lines as in [9].

The proof can be given by using the same process as in the proof of Theorem 3.1.

Since the proof can be given for the space (B(mathcal{C}_{bp})) in a similar way, we omit the details.

Since the proof can be given in the same way for the other spaces, we only show here that (B(mathcal{M}_{u})) is linearly isomorphic to the space (mathcal{M}_{u}).

We will only prove the result for the operator (mathcal{O}(mathcal{T}_{vec{b}})) since a similar proof can be given for the operator (mathcal{V}_{rho}(mathcal {T}_{vec{b}})).

A shorter proof can be given using the advanced definition that a domain G in ({ mathbb C}) is simply connected if every curve in G is contractible in G, or equivalently, if for every Jordan curve J in G, the interior of J belongs to G.

More recently, Curtis Franks takes up the issue of the second incompleteness theorem for weak arithmetic theories in his 2009, challenging Pudláak's argument that an intensionally adequate proof predicate can be given for Q.